Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #17367 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C210 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C22×C22⋊C4 |
| commutator | C23 | C24×C4 |
| Frattini, cosocle | C22×C4 | C25 |
| socle | C22 | C23×C22⋊C4 |
| prime radical | C23×#41 | C22 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
4622 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 36 points:
(2,4)(5,6,10,16,11,15,9,14)(7,12,20,28,18,26,19,27)(8,21,22,30,23,13,24,29)(17,31,33,36,32,25,34,35);(1,2,3,4)(5,7)(6,12)(8,17)(9,19)(10,20)(11,18)(13,25)(14,27)(15,26)(16,28)(21,31)(22,33)(23,32)(24,34)(29,35)(30,36);(5,8)(6,13)(7,17)(9,22)(10,24)(11,23)(12,25)(14,29)(15,21)(16,30)(18,32)(19,33)(20,34)(26,31)(27,35)(28,36);(7,18)(12,26)(17,32)(19,20)(25,31)(27,28)(33,34)(35,36);(5,9,11,10)(6,14,15,16)(7,19,18,20)(8,22,23,24)(12,27,26,28)(13,29,21,30)(17,33,32,34)(25,35,31,36)Polycyclic presentation:
2^9|a^2=h,b^2=f,e^2=i,h^2=i,b^a=bf,c^a=cg,g^a=gi,d^b=di,h^c=hi116 conjugacy classes — view the full character table.
| maximal subgroup of | #72311298, #72311324, #72311482, … |
| maximal normal subgroup of | #72311298, #72311324, #72311482, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6277977);